3.9  Concluding Discussion

Integral equation methods such as the boundary element method are becoming increasingly popular as methods for the numerical solution of linear elliptic partial differential equations such as the Laplace equation. The application of (discrete) collocation to the integral equation formulation requires the computation of the discrete operators. Fortran subroutines for the evaluation of the discrete Laplace integral operators resulting from the use of constant elements and the most simple boundary approximation to two-dimensional, three-dimensional and axisymmetric problems have been described and demonstrated. The computational cost of a subroutine call is roughly proportional to the number of points in the quadrature rule. Ideally, the quadrature rule should have as few points as possible but must still give a sufficiently accurate approximation to the discrete operator. The efficiency of the overall method will generally be enhanced by varying the quadrature rule with the distance between the point p and the element, the size of the element and the wavenumber. The use of the subroutines would be improved by a method for the automatic selection of the quadrature rule so that the discrete operator can be computed with the minimum number of quadrature points for some predetermined accuracy.

The subroutines have been designed to be easy-to-use, flexible, reliable and efficient. It is the intention that the subroutines are to be used as a `black box' which can be utilised either for further analysis of integral equation methods or in software for the solution of practical physical problems which are governed by the Laplace equation.

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